NCERT Solutions for Class 12 Maths Chapter 10 – Vector Algebra Ex 10.2
Page No 440:
Question 1:
Compute the magnitude of the following vectors:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7476/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_31caa58b.gif)
Answer:
The given vectors are:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7476/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_59400371.gif)
Question 2:
Write two different vectors having same magnitude.
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7479/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m77a7631e.gif)
Hence,
are two different vectors having the same magnitude. The vectors are different because they have different directions.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7479/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_96f7d8b.gif)
Question 3:
Write two different vectors having same direction.
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7481/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m19981f5.gif)
The direction cosines of
are the same. Hence, the two vectors have the same direction.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7481/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_2b260d4b.gif)
Question 4:
Find the values of x and y so that the vectors
are equal
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7503/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_1e09f09f.gif)
Answer:
The two vectors
will be equal if their corresponding components are equal.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7503/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_1e09f09f.gif)
Hence, the required values of x and y are 2 and 3 respectively.
Question 5:
Find the scalar and vector components of the vector with initial point (2, 1) and terminal point (–5, 7).
Answer:
The vector with the initial point P (2, 1) and terminal point Q (–5, 7) can be given by,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7505/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m3e8b1d95.gif)
Hence, the required scalar components are –7 and 6 while the vector components are ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7505/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m7fe85b5f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7505/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m7fe85b5f.gif)
Question 6:
Find the sum of the vectors
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7506/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m30282b9b.gif)
Answer:
The given vectors are
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7506/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m30282b9b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7506/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_1573fddd.gif)
Question 7:
Find the unit vector in the direction of the vector
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7508/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m2f1bc4fe.gif)
Answer:
The unit vector
in the direction of vector
is given by
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7508/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m4b61e9ff.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7508/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m2f1bc4fe.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7508/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m8220397.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7508/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_775fd640.gif)
Question 8:
Find the unit vector in the direction of vector
, where P and Q are the points
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7509/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_37573ad.gif)
(1, 2, 3) and (4, 5, 6), respectively.
Answer:
The given points are P (1, 2, 3) and Q (4, 5, 6).
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7509/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m240d73d1.gif)
Hence, the unit vector in the direction of
is
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7509/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_37573ad.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7509/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m66bb6ed6.gif)
Question 9:
For given vectors,
and
, find the unit vector in the direction of the vector ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7511/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m7afd8463.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7511/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m79ed765d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7511/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m3fc656e2.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7511/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m7afd8463.gif)
Answer:
The given vectors are
and
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7511/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m79ed765d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7511/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m3fc656e2.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7511/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m21d9071e.gif)
Hence, the unit vector in the direction of
is
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7511/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_73889a02.gif)
a→+b→a→+b→=i^+k^2=12i⏜+12k⏜.
Question 10:
Find a vector in the direction of vector
which has magnitude 8 units.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7513/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m72569e2.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7513/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_24386c6e.gif)
Hence, the vector in the direction of vector
which has magnitude 8 units is given by,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7513/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m72569e2.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7513/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_b5d998e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7513/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m649daf97.gif)
Question 11:
Show that the vectors
are collinear.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7514/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m45d9d2dc.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7514/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m5bf8a3dd.gif)
Hence, the given vectors are collinear.
Question 12:
Find the direction cosines of the vector ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7516/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_1dda74be.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7516/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_1dda74be.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7516/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_mf23f50e.gif)
Hence, the direction cosines of ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7516/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_3cb64f1d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7516/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_3cb64f1d.gif)
Question 13:
Find the direction cosines of the vector joining the points A (1, 2, –3) and
B (–1, –2, 1) directed from A to B.
Answer:
The given points are A (1, 2, –3) and B (–1, –2, 1).
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7518/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_3c8894e3.gif)
Hence, the direction cosines of
are ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7518/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m439a355c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7518/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m27be3bd4.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7518/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m439a355c.gif)
Question 14:
Show that the vector
is equally inclined to the axes OX, OY, and OZ.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7519/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_26020955.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7519/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_2f6cde44.gif)
Therefore, the direction cosines of ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7519/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_md243c2d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7519/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_md243c2d.gif)
Now, let α, β, and γbe the angles formed by
with the positive directions of x, y, and z axes.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7519/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_4f2bfef5.gif)
Then, we have![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7519/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m2aa8c3f7.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7519/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m2aa8c3f7.gif)
Hence, the given vector is equally inclined to axes OX, OY, and OZ.
Question 15:
Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are
respectively, in the ration 2:1
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7522/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_mc93ff0a.gif)
(i) internally
(ii) externally
Answer:
The position vector of point R dividing the line segment joining two points
P and Q in the ratio m: n is given by:
- Internally:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7522/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_b9eaa80.gif)
- Externally:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7522/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m30020ea7.gif)
Position vectors of P and Q are given as:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7522/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_37b0a2c3.gif)
(i) The position vector of point R which divides the line joining two points P and Q internally in the ratio 2:1 is given by,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7522/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m20bb3b45.gif)
(ii) The position vector of point R which divides the line joining two points P and Q externally in the ratio 2:1 is given by,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7522/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_19a3f4d7.gif)
Page No 441:
Question 16:
Find the position vector of the mid point of the vector joining the points P (2, 3, 4) and Q (4, 1, – 2).
Answer:
The position vector of mid-point R of the vector joining points P (2, 3, 4) and Q (4, 1, – 2) is given by,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7524/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_7187c3d4.gif)
Question 17:
Show that the points A, B and C with position vectors,
,
respectively form the vertices of a right angled triangle.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7526/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_4abb5e40.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7526/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_79bc6ffe.gif)
Answer:
Position vectors of points A, B, and C are respectively given as:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7526/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_mcc60c.gif)
![](https://img-nm.mnimgs.com/img/study_content/content_ck_images/images/67368736482.png)
AB→2+CA→2=35+6=41=BC→2Hence, ABC is a right-angled triangle.
Question 18:
In triangle ABC which of the following is not true:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7528/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_5e1b4db4.jpg)
A. ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7528/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m2a9ed7f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7528/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m2a9ed7f.gif)
B. ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7528/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_2edf6a76.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7528/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_2edf6a76.gif)
C. ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7528/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_mb64d277.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7528/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_mb64d277.gif)
D. ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7528/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_23234a70.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7528/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_23234a70.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7528/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_5e1b4db4.jpg)
On applying the triangle law of addition in the given triangle, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7528/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m7b77e624.gif)
From equations (1) and (3), we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7528/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m7a819ddd.gif)
Hence, the equation given in alternative C is incorrect.
The correct answer is C.
Question 19:
If
are two collinear vectors, then which of the following are incorrect:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7530/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_469c0fd8.gif)
A.
, for some scalar λ
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7530/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_354a54cd.gif)
B. ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7530/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m626eecdc.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7530/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m626eecdc.gif)
C. the respective components of
are proportional
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7530/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_469c0fd8.gif)
D. both the vectors
have same direction, but different magnitudes
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7530/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_469c0fd8.gif)
Answer:
If
are two collinear vectors, then they are parallel.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7530/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_469c0fd8.gif)
Therefore, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7530/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_354a54cd.gif)
If λ = ±1, then
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7530/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_m626eecdc.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7530/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_34d524ff.gif)
Thus, the respective components of
are proportional.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7530/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_469c0fd8.gif)
However, vectors
can have different directions.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/239/7530/NCERT_10-11-08_Khushboo_12_Math_Ex-10.1_5.doc_SG_html_469c0fd8.gif)
Hence, the statement given in D is incorrect.
The correct answer is D.